This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem predon

Description: The predecessor of an ordinal under _E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011) (Proof shortened by BJ, 16-Oct-2024)

		Ref	Expression
	Assertion	predon	$⊢ A \in On \to Pred (E, On, A) = A$

Proof

Step	Hyp	Ref	Expression
1		tron	$⊢ Tr On$
2		trpred	$⊢ (Tr On \land A \in On) \to Pred (E, On, A) = A$
3	1 2	mpan	$⊢ A \in On \to Pred (E, On, A) = A$